Volatility (finance) - Mathematical Definition

Mathematical Definition

The annualized volatility σ is the standard deviation of the instrument's yearly logarithmic returns.

The generalized volatility σ_T for time horizon T in years is expressed as:

Therefore, if the daily logarithmic returns of a stock have a standard deviation of σ_SD and the time period of returns is P, the annualized volatility is

A common assumption is that P = 1/252 (there are 252 trading days in any given year). Then, if σ_SD = 0.01 the annualized volatility is

The monthly volatility (i.e., T = 1/12 of a year) would be

The formula used above to convert returns or volatility measures from one time period to another assume a particular underlying model or process. These formulas are accurate extrapolations of a random walk, or Wiener process, whose steps have finite variance. However, more generally, for natural stochastic processes, the precise relationship between volatility measures for different time periods is more complicated. Some use the Lévy stability exponent α to extrapolate natural processes:

If α = 2 you get the Wiener process scaling relation, but some people believe α < 2 for financial activities such as stocks, indexes and so on. This was discovered by Benoît Mandelbrot, who looked at cotton prices and found that they followed a Lévy alpha-stable distribution with α = 1.7. (See New Scientist, 19 April 1997.)